Sequences of Functions

Problem 1
Let $\{g_n\}$ be a sequence of real valued functions such that $g_{n+1}(x)\leq g_n(x)$ for each $x$ in T and for every $n=1,2,...$. If $\{g_n\}$ is uniformly bounded on T and if $\sum f_n(x)$ converges uniformly on T, then $\sum f_n(x)g_n(x)$ also converges uniformly on T.

Problem 2
Let ${a_n}$ be a decreasing sequence of positive terms. Prove that the series $\sum a_n \sin nx$ converges uniformly on $\mathbb{R}$ if, and only if, $na_n\to 0$ as $n\to\infty$

Only if direction is easy so we omit the proof here, we only prove the if direction
Suppose $na_n\to 0$ as $n\to\infty$, let $B_n(x)=\sum\limits_{k=1}^{n}\sin kx$ $\sum\limits_{k=1}^{n}a_k \sin kx=B_n a_{n+1}-\sum\limits_{k=1}^{n}B_k (a_{k+1}-a_k)$ $$\begin{align*} \bigg|\sum\limits_{k=n+1}^{m}a_k \sin kx\bigg|&=\bigg| B_n a_{n+1}-B_m a_{m+1}+\sum\limits_{k=n+1}^{m}B_k (a_{k+1}-a_k)\bigg|\\ &=\bigg| B_n a_{n+1}-B_m a_{m+1}+ B(a_{m+1}-a_{n+1}) \bigg|,\;\;\ \\ &\mbox{where}\;min\{B_{n+1},B_{n+2},...,B_{m}\}\leq B \leq max\{B_{n+1},B_{n+2},...,B_{m}\}\\ &\leq \bigg|(B_n-B) a_{n+1} \bigg|+ \bigg|(B-B_m) a_{m+1} \bigg|\\ &<2Na_{n+1}+2Ma_{m+1}\\ &<\varepsilon \end{align*}$$

problem 1 can be proved in a similar way

Problem 3
Given a power series $\sum_{n=0}^{\infty} a_n z^n$ whose coefficients are related by an equation of the form $$a_n+Aa_{n-1}+Ba_{n-2}=0$$ (n=2,3,...)
Show that for any $x$ for which ther series converges, its sum is $$\frac{a_0+(a_1+Aa_0)x}{1+Ax+Bx^2}$$

Problem 4
Show that the binomial series $(1+x)^{\alpha}=\sum_{n=0}^{\infty} {\alpha \choose k} x^n$ exhibits the following behavior at the points $x=\pm 1$
a) If $x=-1$, the series converges for $\alpha\geq 0$ and diverges for $\alpha<0$
b) If $x=1$, the series diverges for $\alpha \leq -1$, converges conditionally for $\alpha$ in the inteval $-1<\alpha<0$, and converges absolutely for $\alpha\geq0$